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<h1>Dynamic Response Functions</h1>

<p>We define a dynamic correlation function as
<div class="math">
χ<sub>AB</sub>(τ) = -(1/ℏ)⟨A(τ)B(0)⟩,
</div>
where <span class="math">A</span> and 
<span class="math">B</span> are operators.
</p>

<h2>Density-density response</h2>
<p>
Since <span class="pi">pi</span> 
uses a position basis, we often collect density fluctuations in 
real space.
However, most textbook descriptions
of density fluctioans are in k-space, and results for
homogeneous systems are often best represented in k-space. Here
we give a brief summary of common definitions for pedagogical purposes.
For simplicity we write all formulas for spinless particles.
</p>
<p>
The dimensionsless Fourier transform of the density operator
is (Eqs. 1.11 and 1.66 of Giuliani and Vignale)
<div class="math">
n<sub><b>q</b></sub> 
&nbsp;=&nbsp; Σ<sub>j</sub>
e<sup>-i<b>k</b>·<b>r</b><sub>j</sub></sup>
&nbsp;=&nbsp; Σ<sub><b>k</b></sub>
  a<sup>†</sup><sub><b>k</b>-<b>q</b></sub> a<sub><b>k</b></sub>.
</div>
Note that <span class="math">n<sub>0</sub>=N</span>, the total
number of particles.
To get back to real-space density use
<div class="math">
n(<b>r</b>) = (1/V) Σ<sub><b>q</b></sub>
  n<sub><b>q</b></sub> e<sup>i<b>q</b>·<b>r</b></sup>.
</div>
</p>
<p>
For each of these, we define frequency-dependent density operators,
<div class="math">
n(<b>r</b>,iω<sub>n</sub>) =
  ∫<sub>0</sub><sup>βħ</sup>&nbsp; n(<b>r</b>,τ) e<sup>iω<sub>n</sub>τ</sup> dτ,
</div>
and
<div class="math">
n<sub><b>q</b></sub>(iω<sub>n</sub>) =
  ∫<sub>0</sub><sup>βħ</sup>&nbsp; n<sub><b>q</b></sub>(τ) e<sup>iω<sub>n</sub>τ</sup> dτ,
</div>
where <span class="math">iω<sub>n</sub> = 2πinkT/ħ</span>
are the Matsubara frequencies.
Within the <span class="pi">pi</span> code, these frequency-dependent
densities are easily calculated
with fast Fourier transforms, which are most efficient when the number of
slices is a power of two.
</p>

<h3>Real-space response</h3>
<p>The imaginary-frequency 
response of the density to an external perturbation is
given by (Ch 3.3 of Guiliani and Vignale),
<div class="math">
δn(<b>r</b>,iω<sub>n</sub>)
 = ∫ d<b>r</b> χ<sub>nn</sub>(<b>r</b>,<b>r</b>', iω<sub>n</sub>) 
  V<sub>ext</sub>(<b>r</b>', iω<sub>n</sub>).
</div>
In k-space this takes the convienent form,
<div class="math">
δn(<b>q</b>,iω<sub>n</sub>) =  Σ<sub><b>q</b>'</sub>
χ<sub>nn</sub>(<b>q</b>,<b>q</b>',iω<sub>n</sub>) V<sub>ext</sub>(<b>q</b>',iω<sub>n</sub>).
</div>
where the external potential in k-space satisfies
<div class="math">
V<sub>ext</sub>(<b>r</b>') = (1/V) Σ <sub><b>q</b>'</sub>
  V<sub>ext</sub>(<b>q</b>') e<sup>i<b>q</b>'·<b>r</b>'</sup>,
</div>
and
<div class="math">
V<sub>ext</sub>(<b>q</b>') 
  = ∫ d<b>q</b>' e<sup>-i<b>q</b>'·<b>r</b>'</sup> V<sub>ext</sub>(<b>r</b>').
</div>
</p>
<p>
These response functions are related to imaginary-frequency 
dynamic correlation functions,
<div class="math">
χ<sub>nn</sub>(<b>r</b>,<b>r</b>',iω<sub>n</sub>)
= -(1/βħ²)〈n(<b>r</b>,iω<sub>n</sub>) n(<b>r</b>',-iω<sub>n</sub>)〉,
</div>
and
<div class="math">
χ<sub>nn</sub>(<b>q</b>,<b>q</b>',iω<sub>n</sub>)
= -(1/βħ²V)〈n<sub><b>q</b></sub>(iω<sub>n</sub>) 
             n<sub>-<b>q</b>'</sub>(-iω<sub>n</sub>)〉.
</div>
</p>
For a homogeneous system, 
<div class="math">
χ<sub>nn</sub>(<b>q</b>,<b>q</b>',iω<sub>n</sub>)
= -(1/βħ²V)〈n<sub><b>q</b></sub>(iω<sub>n</sub>)
             n<sub>-<b>q</b></sub>(-iω<sub>n</sub>)〉
δ<sub><b>qq</b>'</sub>.
</div>

<h3>Structure factor</h3>

<p>
The dynamic structure factor
<span class="math">S(<b>k</b>,iω<sub>n</sub>)</span>
measures the density response of the system,
<div class="math">
S(<b>k</b>,iω<sub>n</sub>) = -(V/ħN) 
χ<sub>nn</sub>(<b>k</b>,<b>k</b>,iω<sub>n</sub>)
</div>
The static structure factor is defined for equal time, not for
<span class="math">ω<sub>n</sub> → 0</span>,
<div class="math">
S(<b>k</b>) = (1/N) 
〈n<sub><b>k</b></sub>(τ=0) n<sub>-<b>k</b></sub>(τ=0)〉.
</div>
In terms of <span class="math">χ<sub>nn</sub>(<b>q</b>,<b>q</b>',iω)</span>,
the static structure factor is given by (<em>prefactor is wrong</em>)
<div class="math">
S(<b>k</b>) = -(V/ħN) Σ<sub>ω<sub>n</sub></sub> 
χ<sub>nn</sub>(<b>k</b>,<b>k</b>,iω<sub>n</sub>)
   e<sup>-iω<sub>n</sub>τ</sup>.
</div>
</p>
<h3>Polarizability</h3>

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